Pyöritysmatriisin

Pyöritysmatriisin, also known as a rotation matrix, is a fundamental concept in linear algebra and geometry. It is a square matrix used to perform a rotation in a Euclidean space. The most common rotation matrices are 2x2 and 3x3 matrices, which correspond to rotations in the plane (2D) and in three-dimensional space (3D), respectively.

In 2D, a rotation matrix R(θ) for an angle θ is defined as:

R(θ) = [cos(θ) -sin(θ)]

[sin(θ) cos(θ)]

This matrix can be used to rotate a vector (x, y) counterclockwise by an angle θ. The resulting vector (x', y') is given by the matrix multiplication:

(x', y') = R(θ) * (x, y)

In 3D, rotation matrices are more complex and can be defined for rotations around the x, y, or z axes. For example, a rotation matrix Rz(θ) around the z-axis is:

Rz(θ) = [cos(θ) -sin(θ) 0]

[sin(θ) cos(θ) 0]

[ 0 0 1]

Rotation matrices have several important properties. They are orthogonal, meaning their transpose is equal to their inverse, and their determinant is 1, indicating that they preserve the volume of the space they operate on. Additionally, the product of two rotation matrices is also a rotation matrix, which allows for the composition of multiple rotations.

Pyöritysmatriisin is widely used in computer graphics, robotics, and other fields where rotations are a common operation. Its simplicity and efficiency make it a valuable tool for transforming and manipulating geometric data.